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\author{五六七 }
\title{从开普勒行星运动定律到万有引力定律 }

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\begin{document}

\maketitle

\begin{abstract}
由开普勒的三个行星运动定律和牛顿第二运动定律，通过微积分得出万有引力公式。
\end{abstract}

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\section{问题描述}


开普勒等人在很多观察数据的基础上，总结出了行星运动的三大定律。牛顿是如何由此得出他的著名的万有引力定律的？
参考 \cite{jiangqiyuan} 第176页。

\begin{figure}[!ht]
\centering
\includegraphics [height=6cm,width=8cm]{ellipse_orbit.png}
\caption{行星沿椭圆轨道运行}
\end{figure}

\section{模型假设}
\begin{enumerate}
\item[K1:] 太阳系中各个行星的运行轨道是椭圆，太阳位于其中的一个焦点。
\item[K2:] 对每个行星，在同样时间内``太阳-行星''这个半径扫过同样的面积。
\item[K3:] 对每个行星，运行周期的平方与长半轴成正比。
\item[N2:] 物体运动的加速度等于其受到的力除以其质量。
\end{enumerate}

%\underline{测验A}:

\section{模型解答}
\subsection{极坐标，单位正交标架}
取如图所示的极坐标，其中太阳是原点，记太阳到行星的向量为 $\vec{\bf r}$, 其大小为 $r$. 记从长半轴逆时针转到该向量的角度为 $\theta$. 这两个是与时间有关的变量，它们是时间 $t$ 的函数。分别记长半轴和短半轴为 $a$ 和 $b$, 记 $c=\sqrt{a^2-b^2}$.
在这样的记号下，向量 $\vec{\bf r}$ 与极坐标 $(r,\theta)$ 有着相同的信息。

记在 $(r,\theta)$ 这一点的{\bf 极坐标单位正交标架}为 $(\vec{\bf r}_u,\vec{\pmb\theta}_u)$. 即有
\begin{eqnarray}
\vec{{\bf r}}_u     &=& \cos\theta \vec{{\bf i}} + \sin\theta\vec{{\bf j}}, \\
\vec{\pmb{\theta}}_u &=& -\sin\theta \vec{{\bf i}} + \cos\theta\vec{{\bf j}},
\end{eqnarray}
其中 $(\vec{{\bf i}},\vec{{\bf j}})$ 是{\bf 直角坐标单位正交标架}，在不同地方都是一样的。
而 $(\vec{\bf r}_u,\vec{\pmb\theta}_u)$ 是时间的函数，即在行星运行到不同的位置时，它们的值是不一样的。


\underline{测验A}：计算验证极坐标单位正交标架关于时间 $t$ 的导数满足下述等式。
\begin{eqnarray}
\vec{{\bf r}}_u\,'(t)    &=& \theta'(t) \cdot \vec{{\bf r}}_\theta (t)   \\
\vec{{\bf r}}_\theta\,'(t)    &=& -\theta'(t) \cdot \vec{{\bf r}}_u (t)
\end{eqnarray}


\subsection{椭圆的方程}

\underline{测验B}：设一焦点为原点，验证椭圆的极坐标方程可以写成
\begin{eqnarray}
r = \frac{b^2}{a+c\cos\theta},\,\,\, 0\le \theta\le 2\pi.
\end{eqnarray}



\subsection{把假设数学化}
定义了上述变量之后，可以把开普勒的三个定律和牛顿第二定律写成这样。
\begin{enumerate}
\item[K1:] 得到上述椭圆的方程。
\item[K2:] $\frac{1}{2} r(t)^2 \theta'(t) = A $ 是一个常数（与行星有关）。
\item[K3:] $T^2 = ka^3$, 其中 $k$ 是一个常数（与太阳有关）。
\item[N2:] $\vec{\bf f} = m \vec{\textbf a} $, 而 $\vec{\textbf a}(t) = \vec{\bf r}\,''(t)$.
\end{enumerate}

本文的目标是由此得出万有引力公式
\begin{eqnarray}
\vec{\bf f} = -k\frac{4\pi^2 m}{\lambda r^2} \vec{{\bf r}}_u.
\end{eqnarray}


\subsection{证明思路}

计算出向量值函数 $\vec{{\bf r}}\,(t)$ 的二阶导数 $\vec{{\bf r}}\, ''(t)$, 然后代入牛顿第二定律。
\begin{eqnarray}
\vec{{\bf r}}\,(t)    &=& r(t) \vec{{\bf r}}_u (t)   \\
\vec{{\bf r}}\,'(t)    &=& \\
\vec{{\bf r}}\, ''(t)  &=&
\end{eqnarray}

\underline{测验C}：完成这个计算。



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\section{编程计算}





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\section{回答问题}




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%\section{参考文献 }
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\bibitem{jiangqiyuan} 姜启源, 谢金星, 叶俊. \emph{数学模型}, 高等教育出版社. 2018年5月第5版. （176页，第5.9节）
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\end{document}

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\subsection{内容提要}
\begin{itemize}
\item 从开普勒三个定律推导出万有引力定律。
\item 从开普勒第二定律推导出引力与距离的平方成反比例。
\end{itemize}

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\subsection{相关问题}
\begin{enumerate}
\item 什么是开普勒的行星运动三定律？
\item 什么是牛顿的运动三定律？
\item 什么是万有引力定律？
\item 用微积分的方法，从开普勒三大定律得出牛顿的万有引力定律。
\item 用几何作图的方法，从开普勒第二定律得出万有引力与距离的平方成反比例。
%\item 椭圆与双曲线的
%\item 
%\item 
\end{enumerate}

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